# Cutting a Möbius strip

Let's do a little introduction to topology recreationally, by a set involving a tape or banda de Moebius.

It's an exercise I do in the first geometry classes that I give my students at UPM aeronautical and used to explore basic concepts while stimulating the curiosity for science.

The Moebius band Moebius on tape (pronounced / møbiʊs / or Spanish often “moebius”) is a surface with one side and one edge, the contour component. It has the mathematical property of being an object nonorientable. It is also a ruled surface. He co-discovered independently by the German mathematicians August Ferdinand Möbius and Johann Benedict Listing in 1858. (W)

"The book of mathematics. From Pythagoras to the 57th dimension. 250 milestones in the history of mathematics ", Clifford A. Pickover, Bookseller, 2011.

The Topology is the study of those properties of solids that remain unchanged by continuous transformations. It is a mathematical discipline that studies the properties of topological spaces and continuous functions.

The Topology interested in concepts such as proximity, Number of holes, the kind of consistency (the texture) presenting an object, compare and classify objects… (W)

For activity need only the elements you see in the picture:

Sheet of paper

Lápiz

Scissors

Tape

This activity serves to motivate students, both to stimulate thought and rational analysis.

Can be performed in a short time, half an hour, being an elapsed time that provides a high intellectual performance

# Build bands

First we build two bands with two strips of paper, one in a ring and the Moebius strip.

Will cut a rectangular strip of paper and put a bit of tape on one end.

The idea is to merge the two shorter edges of the rectangle to form a circular band.

We can do this in two ways, as we want a normal band or the Möbius strip.

First we will make a normal band. Paper will join the ends of its short side to obtain a cylindrical shape, ring.

This surface has two sides, an inner and outer.

If we go one side with a finger, never would play the other side.

Then build the banda de Moebius. In this case when the ends are glued we turned to one.

This makes rotation of the paper we connect the two sides of the surface; obtains a surface with a single, and that if we repeated the above operation, scouring the surface with a finger, would spend the whole surface.

This idea allows us to speak of a number of faces peer (2) the odd (1).

We built two different bands that will serve to “play” with them and stimulate basic analysis that allows us to work further with the abstract surfaces.

# Surfaces of the bands

The elements necessary to start scanning are ready. Review the number of faces while the band prepared to be cut.

With the pencil will draw, from any point, scroll one line up for their central. continue drawing until you close the line to complete the return to the band.

We divided the band by a line “equidistant” of its ends. We say that this line is away 1/2 ( medium).

While in the normal range only draw half of the surface (face which we started), in the Moebius strip will line the entire surface, the only face that there.

You can also call “Midline” of the face.

As a further exercise, We can draw lines at other distances:instead of dividing the width into two parts, we will do three, four …

It leaves open the exercise to motivate the exploration of the year, raising some questions:

• If divided into three parts, to draw the lines in the band Can we do it without lifting the pencil from the paper? namely, with a single stroke we cross the tape, completing lines.
• By building the band, if instead of rotate once, turn two, three, four…. ?How many faces have surface?

# Cut surfaces

The most interesting part of the game comes when “cut” tape along the line we have previously marked. Before you begin to cut, Are we able to predict what the va pass?

We begin with the tape “normal”, one that has no turning.

We will follow the line drawn to return to the point where we have begun to cut.

Will the same result with the other tape?

¿Will have one of the tapes as a result?

¿From one of the faces in each case?

Anticipation is an interesting response to the analysis engine. We see that the ribbon cutting get two exactly equal to the original, except in width, has been halved.

What will happen to cut the ribbon moebius?

We see that in this other case it is obtained also is a half band width than the original, but “only a tape”.

Its length has become twice the primitive, After the “just had a face” !!!

How many faces does the new band?

This exercise does not end here, Now we try to generalize the results in case we mark lines rather than in the midline, a third of the distance (width), or a quarter, the fifth one …..

We can also speculate on what would happen if we make two turns to build the tape, the three, or four….

We can build a few tapes to experiment and draw conclusions, the result may be surprising.